"A good stock of examples, as large as possible, is indispensable for a thorough understanding of any concept, and when I want to learn something new, I make it my first job to build one." – Paul Halmos

Archive for the ‘math.GR’ Category

Previously we learned how to count the finite index subgroups of the modular group . The worst thing about that post was that it didn’t include any pictures of these subgroups. Today we’ll fix that.

The pictures in this post can be interpreted in at least two ways. On the one hand, they are graphs of groups in the sense of Bass-Serre theory, and on the other hand, they are also dessin d’enfants (for the rest of this post abbreviated to “dessins”) in the sense of Grothendieck. But you don’t need to know that to draw and appreciate them.

Previously we learned how to count the number of finite index subgroups of a finitely generated group . But for various purposes we might instead want to count conjugacy classes of finite index subgroups, e.g. if we wanted to count isomorphism classes of connected covers of a connected space with fundamental group .

There is also a generating function we can write down that addresses this question, although it gives the answer less directly. It can be derived starting from the following construction. If is a groupoid, then , the free loop space or inertia groupoidof , is the groupoid of maps , where is the groupoid with one object and automorphism group . Explicitly, this groupoid has

objects given by automorphisms of the objects , and

morphisms given by morphisms in such that

.

It’s not hard to see that , so to understand this construction for arbitrary groupoids it’s enough to understand it for connected groupoids, or (up to equivalence) for groupoids with a single object and automorphism group . In this case, is the groupoid with objects the elements of and morphisms given by conjugation by elements of ; equivalently, it is the homotopy quotient or action groupoid of the action of on itself by conjugation.

In particular, when is finite, this quotient always has groupoid cardinality . Hence:

Observation: If is an essentially finite groupoid (equivalent to a groupoid with finitely many objects and morphisms), then the groupoid cardinality of is the number of isomorphism classes of objects in .

Yesterday we described how a (finite-dimensional) projective representation of a group functorially gives rise to a -linear action of on such that the Schur class classifies this action.

Today we’ll go in the other direction. Given an action of on explicitly described by a 2-cocycle , we’ll recover the category of -projective representations, or equivalently the category of modules over the twisted group algebra, by taking the homotopy fixed points of this action. We’ll end with another puzzle.

Today we’ll resolve half the puzzle of why the cohomology group appears both when classifying projective representations of a group over a field and when classifying -linear actions of on the category of -vector spaces by describing a functor from the former to the latter.

Three days ago we stated the following puzzle: we can compute that isomorphism classes of -linear actions of a group on the category of vector spaces over a field correspond to elements of the cohomology group

.

This is the same group that appears in the classification of projective representations of over , and we asked whether this was a coincidence.

Before answering the puzzle, in this post we’ll provide some relevant background information on projective representations.

Previously we described what it means for a group to act on a category (although we needed to slightly correct our initial definition). Today, as the next step in our attempt to understand Galois descent, we’ll describe what the fixed points of such a group action are.

John Baez likes to describe (vertical) categorification as replacing equalities with isomorphisms, which we saw on full display in the previous post: we replaced the equality with isomorphisms , and as a result we found 2-cocycles lurking in this story.

I prefer to describe categorification as replacing properties with structures, in the nLab sense. That is, the real import of what we just did is to replace the property (of a function between groups, say) that with the structure of a family of isomorphisms between and . The use of the term “structure” emphasizes, as we also saw in the previous post, that unlike properties, structures need not be unique.

Accordingly, it’s not surprising that being a fixed point of a group action on a category is also a structure and not a property. Suppose is a group action as in the previous post, and is an object. The structure of a fixed point, or more precisely a homotopy fixed point, is the data of a family of isomorphisms

which satisfy the compatibility condition that the two composites

and

are equal, as well as the unit condition that

where is the unit isomorphism . This is, in a sense we’ll make precise below, a 1-cocycle condition, but this time with nontrivial (local) coefficients.

Curiously, when the action is trivial (meaning both that and that ), this reduces to the definition of a group action of on in the usual sense. In general, we can think of homotopy fixed point structure as a “twisted” version of a group action on where the twist is provided by the group action on .

Yesterday we decided that it might be interesting to describe various categories as “fixed points” of Galois actions on various other categories, whatever that means: for example, perhaps real Lie algebras are the “fixed points” of a Galois action on complex Lie algebras. To formalize this we need a notion of group actions on categories and fixed points of such group actions.

So let be a group and be a category. For starters, we should probably ask for a functor for each . Next, we might naively ask for an equality of functors

but this is too strict: functors themselves live in a category (of functors and natural transformations), and so we should instead ask for natural isomorphisms

.

These natural isomorphisms should further satisfy the following compatibility condition: there are two ways to use them to write down an isomorphism , and these should agree. More explicitly, the composite

should be equal to the composite

.

(There’s also some stuff going on with units which I believe we can ignore here. I think we can just require that on the nose and nothing will go too horribly wrong.)

These natural isomorphisms can be regarded as a natural generalization of 2-cocycles, and the condition above as a natural generalization of a cocycle condition. Below the fold we’ll describe this and other aspects of this definition in more detail, and we’ll end with two puzzles about the relationship between this story and group cohomology.